I’ve had that conversation more times than I count, most recently last week. Are you going to fail? Can’t answer that, although if you’re asking there’s actually a good chance you won’t if you’re asking this question in April.

Here’s the secret I can’t tell you when you’re a student: some of your classmates who look really confident have no idea what they’re doing.

Some of your classmates who you wouldn’t imagine are acing the class.

That one guy who stays after to ask questions all the time? He’s so intimidating because you don’t even understand the questions but you’re sure they’re super-advanced? He’s got some good ideas but I don’t understand half his questions either, because he doesn’t know how to translate the ideas he’s got from, “I feeeel like this should work” to actual mathematics.

The difference between you? He just assumes his wacky feeeelings about mathematics should translate to something real, while you assume you’re probably wrong.

In high school, math is often pretty cut and dried. Sometimes there’s a specific process that your teacher wants to use to solve a problem; it’s in the curriculum so that’s how you’re graded. Specific topics are covered — you have to get through the curriculum — there’s a standardized test coming up.

In college, if you’ve got professors for teachers they’ve been through a PhD. They all struggled with some previously unsolved math problem and maybe they still do. Creativity for problem-solving is desired in a different way. I promise you, as a college prof, that if you use some other method to solve the problem *and you can prove you’re right *then I’ll give you credit. A lot of math breakthroughs happen because a mathematician said, “What would have to be true to make this true? What would happen if I did this illegal move in mathematics and just… tried to make things work?”

Differentiating at a cusp? Illegal, right? “What if I do it anyway?” Get distributions, generalized functions. Take the square root of a negative number: impossible! “What if I do it anyway?” Complex numbers. Divide by zero: “What would it take to make this work?” Limits. You can’t integrate this function… invent Lebesgue integrals!

So, are your classmates smarter than you? Some can do an integral faster but don’t understand the theorems. Some understand the big ideas but are slow at computation. Some want applications to physics and some think applications are really annoying. The ones you think are the smartest probably aren’t, because you are probably confusing confidence with smarts (they look the same if you’re just learning the subject). And the ones who aren’t the smartest might go the farthest. I have a PhD in math now. I was definitely good at it in high school, but I was never The Best. The guys who were The Best? Not doing math now.

Don’t discount yourself.

]]>“I did know all the topics inside and out on the multiple choice…”

“I got so flustered by the free response…”

“Stumped and no idea what to do.”

“A downward spiral from there.”

“I didn’t know where to find the formula so I said f*** this.”

“Looked like absolutely nothing we were supposed to have memorized.”

“It wasn’t anything we were supposed to memorize.”

“After that free response, there’s absolutely no way I can get a 5 now.”

“Now I feel like an idiot.”

Instead you want to be the students saying, “I think they were well-designed questions ‘cause all of them could be simplified to easier types of questions.” You need a few things for the free response question that you don’t need for multiple choice:

- Insight (what’s the easier type of problem?)
- Creativity (how can I transform this?)
- Skills in writing up your solution, so the grader says, “Amazing!” instead of, “What… is this kid trying to say? “

(Yeah, I’ve talked to some AP math graders.)

Want to avoid the shame spiral above?

- Check out my articles on sticking the landing, reverse-engineering the grader’s desires, and finding the problem-solving path.
- Sign up for emails with tips & tricks & solved problems.

At a lot of colleges, yes, AP Calc BC is good enough and you can test out of the first year of college. If you’re going to Caltech or MIT or Harvard or U Michigan or Cornell, it’s not a replacement. Even at the University of Minnesota, where I’ve taught a lot, most freshmen who come in with a 3 on the AP Calc exam aren’t exactly prepared for the next semester. We adjust the curriculum for second semester calc knowing that students who start there due to a 3 on the AP are going to have a little bit of a slower start.

What’s the difference between calc at a tough college and AP Calc? Depth. At a tough college, all of your instructors are going to be working on or already have a PhD. Whatever their personal qualities or teaching ability, they’ve been through a lot of math and they have a lot of depth of knowledge. High school teachers, on the other hand, have been through school for a teaching certificate. All of them have a lot more education in education than college instructors. Their math background, though, varies from “I have a PhD or master’s and wanted to teach high school!” to “I got a math major and it was great!” to “I was supposed to be the French teacher but they couldn’t find anyone to cover this class.” I am not kidding: check out the stats on the percentage of high school math teachers in the US who don’t have a math major or a certificate in math. The most important factor in your success besides your own effort is having a good teacher. Either you have one already, or you need to find one (and I’m volunteering!).

That’s why a lot of you get, “Because I said so,” or “because that’s how they want it written,” as answers to your questions in class in high school. That’s why your high school teacher might care about whether you write 1 1/2 versus 3/2. (FYI we don’t use mixed fractions in college math. They’re useless for computation. Fractions or decimals. That’s it.) College math teachers have their own problems, notably a complete lack of education in how to teach unless they went out and got this training themselves — but they usually know a lot of math.

If you really want to be ready for college math, it might help to get some insight from someone who’s already there. Talk to your friends who have gone on to college already — no one can give quite the same insight as someone who went to your high school and is currently taking college math. But also look further along the road: someone like me, who has taught a lot of college math, can give you a 30,000 foot view of high school math, college math, and math out in the real world.

If you’re planning on taking the AP test this spring, sign up for emails with study help and links to videos on the free response section of the AP calculus exam! And good luck!

]]>A strategy that can help is short-circuiting your anxiety habits by getting back into your body during the test!

Think about what you tend to do when you take a test and don’t do as well as you like, even though you studied hard and thought you knew the material. Of course you’re going to have to look at your pre-test prep honestly to see if you *actually* knew what you were doing, but if that prep is solid and you’re still sabotaging yourself, pick one of the following and try it:

**Deep breathing.**Set a time beforehand to stop what you’re working on and take a few deep breaths while looking straight ahead in the testing room (not down at your paper). If it’s a 2.5 hour test, for instance, work for one hour then take five deep, slow breaths, relaxing your hands in your lap and letting go of that pencil. Consciously try to drop all your other thoughts, or think about puppies or sea horses. If neither of those is your style, breathe in counting to 4, hold it for a count of 4, breathe out counting to 4, pause for a count of 4, repeat 4 more times. Now return to your work.**Take a walk.**If it’s allowed, take a “bathroom break” halfway through. (Remember testing etiquette: in general, you have to wait until no one else from the test is using the restroom to have your turn! This avoids the appearance of impropriety/collusion.) Either actually go to the bathroom, or just go out and stretch your legs and arms and get a drink of water. The point is to get up, shake out your physical tension, then go back after a little reset.**Eat a snack.**Ok, let’s be really honest here: my favorite is to bring some chocolate, especially if I’ve been eating a bit of chocolate when studying for the test. Not only does the food give you a bit of energy (and chocolate provides a teeny bit of caffeine), but if you’ve been studying with a certain food or scent, then the return of that food or scent can prompt better recall of the material you were studying! Check if food is allowed in your testing room, though, and don’t bring anything in a noisy wrapper or your fellow students will hate you.**Quick seat stretch.**Stop at a pre-picked time or after you’ve made a first pass through all the problems and do some wrist and arm stretches if you can’t get up or leave. A few unobtrusive ones: clasp your hands together in front of you and rotate them to the right and left; stretch your fingers back on each hand using the other hand; put each arm across the front of your body and pull with the other arm to get at the tension in your shoulders.

What’s the point again? Stop your mental anxiety cycle by refocusing on the body you live in. Get some physical feedback from your senses to pull you off of the mathematical hamster wheel you might be tempted to get on. Remember, on an exam, you want to think *hard enough* but not *too hard: *let your previous studying and hard work carry you through!

Check out our previous posts on stereotype threat and test-taking for more ideas — or suggest your own.

He talks about the people he knows and cares about who do math — or who have quit because of the headwinds they’ve faced. It’s easy to say, “If you have enough passion, you’ll be willing to face any rejection and you’ll keep going,” but that just doesn’t reflect human life. Some people do persevere despite all the odds. Others have to pay bills or set their dreams aside for a time. Francis’s address urges math profs to look out for the students coming up who are facing these headwinds, of all kinds, and help them through. So many factors go into every stage of success that it’s easy to lose track of them. It’s easy to think, “I should be doing better at this. Since I’m not, I’m just not cut out for this.”

Read Francis’s address to be reminded of the fuller story and be encouraged. Yes, it’s addressed to profs, not students, but peek behind the office door and check out what we are talking about!

Play, beauty, truth, justice, love — these are the reasons Francis wants us to do math.

- Reading about role models has been shown to help combat stereotype threat, the secret saboteur in your own mind if you’re a woman or person of color doing math in the US.
- Learning about mathematical role models helps you learn about math! What do these people care about? Why? How did they learn about it? The math you get in school is the “dead butterfly board” of mathematics; your math role models caught live butterflies in the wild!
- Learning a little history helps you combat the erasure of stories and histories that always happens over time. Somehow reading about modern women doing math you get the idea that no woman ever did math before; reading about modern black mathematicians you get the idea that no black people ever did math before. It seems like it’s a big deal — and hard, and maybe too hard. But people from all backgrounds have been very successful in math for hundreds of years. It’s not new, and it’s still wonderful.
- Besides, these are really interesting people!

A few examples: I’m teaching about Taylor series for trig functions right now. Great — they’re named for Brook Taylor (see comment below) who wrote about them in 1715 in England. But even in Britain, it’s a Scottish guy, James Gregory, who actually came up with them. And both these guys were about 300 years after a guy named Madhava of Sangamagrama, an astronomer and mathematician who started the Kerala school of astronomy and math in India. A few hundred years before Europeans got really comfortable with the concept of infinity, he was playing around with series and found series expansions for sine, cosine, and arctangent, as well as a series that gives pi. Why the heck was he doing this in around 1400? Astronomy! There were a whole bunch of mathematicians and astronomers in Kerala working on this, for similar reasons to the Mayans: the urge to understand time and space, coming both from religious and time-keeping reasons.

Madhava was pretty advanced, figuring out series expansions for arctangent in 1400 in India, but he wasn’t alone. Around the same time a guy named Janshid al-Kashi was also approximating pi with series. He grew up near what is today Iran, and then got a great job as lead astronomer and mathematician in Samarkand (in today’s Uzbekistan). Samarkand is one of the oldest cities on the Silk Road and it’s so beautiful…

Europe started dealing with infinite series and calculus in the 1600s — we all know about Newton and Leibniz, and they’ve got really cool stories too. (Much more interesting guys than you might think!) Maria Agnesi wrote the first book covering both differential and integral calculus in 1748. She was Italian and very religious; ended up being a nun and devoting her life to charity after doing math and astronomy for a while. In 1759 Emilie du Chatelet published her translation of and commentary on Newton’s Principia Mathematica. She had also done a lot of work on philosophy, optics, heat, light, and kinetic energy, corresponding with Bernoulli and Euler. At the same time, she had an arranged marriage with some French nobleman, had a (pretty public) affair with the philosopher Voltaire, then fell in love with a poet — and she had three kids. Not only that, she invented a concept pretty close to that of modern financial derivatives when she had to raise a lot of money really fast after losing a fortune to cheating card sharks during one epic night of gambling. Me, I’m impressed and inspired she had time for all that.

For more recent inspiration, you of course can watch the movie “Hidden Figures” about the women who did the computations to get the US to the moon. Women have been instrumental in American and British government math since before World War II (check out the “computers” of Bletchley Park, helping crack Enigma) and made the transition to math with electronic computers really well, helping to establish the basis of computer programming too. The second black woman to get a math PhD in the US is a good example: Evelyn Boyd Granville worked on Project Mercury and Project Vanguard for NASA and then was a senior mathematician at IBM, right at some of the most transitional times in computer technology, before moving to academia. Imagine her motivations — are they so different than Madhava’s or Janshid’s? Or the Mayans? There are always practical motivations — a job that pays well, figuring out the date of the next eclipse or the location of a particular satellite — but what it comes down to for all of us in the end is the curiosity that gets under our skin once we get started on a problem. Maybe you’ll find that in math, even if you don’t expect it, and maybe somewhere else, but it’s one of the things that makes life worth living!

*Brook Taylor is his own cautionary example. The poor man died really early, but the Wikipedia page on him says that he is less famous than he could be because he couldn’t express his ideas through writing “fully and clearly.”*

*photo by Ekrem Canli under https://creativecommons.org/licenses/by-sa/3.0/*

I spend time working on pure math research but also on helping students with their current math homework problems. Honestly, I don’t remember the answers to everything. What do I do, then?

**Draw a picture.**Using the info I’ve got, draw as complete a picture as I can, or a couple. What’s missing from the picture? What could be a bit more general? This works for, “How high does the apple get if it is thrown at 30 degrees and velocity 1.2 m/s?” as well as for, “What is the probability that max(X,Y) is less than .5 if X and Y are the coordinates of a point chosen randomly from the unit square?” In the first problem, my picture would include some representation of gravity and I’d start thinking of critical points or the line of symmetry for a parabola. The second picture might remind me that I need to use area for my calculation.**Look for similar problems in my past.**Obviously this only works directly if you’ve done a similar problem! However, you can use this idea more generally: the probability problem I mentioned above is, once you draw the picture, just a ratio of areas. I’ve computed a lot of areas before. The apple-throwing problem could be like a physics problem, like a precalc problem about the vertex of a parabola, like a critical point problem in calculus… There are a lot of analogies we can make and they often hold a clue to the solution.**Look for similar problems in someone else’s past.**Ask a friend. Read the book for examples. Search the internet for examples. I’m not saying you should offload your work on someone else: I’m saying that judicious use of your other resources is smart. Maybe you won’t find the exact same problem but you’ll find something similar, and then you’ll have the insight to solve your own problem. This often happens in real research: just this week I was reading a friend’s old research paper and saw something that looked familiar, and I was able to say, “Hey, those polynomials are just what I need right now for my own project!” A few weeks ago at a conference I talked to a famous guy in my field and asked how he’d had a new insight, and he said, I was talking to an old friend who is in physics about my problem, and he said it looked just like a system of particles that was really explained in 1989. Yep — that was enough!**Just follow my computational nose, or hands, or whatever.**Seriously, it’s like doing a puzzle: just start writing things down and doing things that are true (adding the same thing to both sides of an equation, taking square roots of everything in sight, multiplying by x, differentiating all the things). Follow the rules of logic, but not in a logical way, and see what happens — just like trying that puzzle piece in every wavy free segment until it fits. Sometimes if you just write you will discover things. Rely on your hands, not your brain, for a few minutes.

Play with it. Juggle the pieces of your problem. Lie on the floor and stare at the ceiling. Get a change of scenery. Eat some chocolate. Try to explain it to someone else. Write stuff down.

Have fun!

]]>Something I hear all the time in my math classes!

Take it from a college professor — YES, you need to write in my math classes!

Communication. Yep, the most clichéd of statements. True for a reason.

- If you become a professional mathematician, you’ll have to write for publication for other mathematicians and you’ll have to write grant applications.
- If you become a chemist, you’ll have to write reports or white papers that explain either the technical or the business side of your work.
- If you become a novelist, you’ll have to plot your novels, and frankly knowing how to write a proof teaches you to think about how to wrap up loose ends. The romance novelist Courtney Milan was a math major and then a law professor, and she’s definitely putting to use those logical writing skills.
- If you go into business, there’s a good chance you’ll need to write to explain decisions to others.

There are plenty of jobs where you won’t have to write, but college does concentrate on preparing you for jobs in which writing is an asset. Chefs and personal trainers and portraitists and cellists and plumbers don’t *need* to write, but many do anyway and see career boosts or alternative revenue streams come their way because of it.

(I thought we were talking about math?)

Why is explaining your math reasoning or proving your result at all related to this? Because in math writing, you are focused on proving — PROVING — that you are right. There can be no argument if you’ve done it well. Your work is clear to the reader and has an air-tight argument. This is very different than writing in economics or history class (there will always be a Marxist or libertarian to argue with you), or writing a literary short story (you want people to find alternate meanings!).

A good skill for someone writing their math problems up is *critically reading their own work*. After you write things down, look for all the problems you can find: unclear bits, leaps of logic, typos, etc. With a poem you wrote about a break-up, you just want to express your feelings and you don’t want to be harsh on yourself. With a math solution, it’s outside you. It’s not about your feelings, it’s a sequence of equations or logical statements with connecting English sentences. This is liberating. Since the math solution isn’t about you, you can rewrite it and argue with it and throw it in the trash and try again all you like! When someone points out a flaw in your argument, you can say, “Yeah, you’re right! … I’ll try to figure out how to fix that!” instead of feeling that they said anything about your self.

Realize that writing up your solution isn’t about showing your thought process, it’s about *proving you are right*. On some level, no one cares how you got to what you think. (On another level, I definitely do care how you got to it!) Again, liberating. If you got super-confused along the way and then figured it out, you don’t need to show that. You just write up your finished reasoning and make it look good.

*Say as little as you can* while saying everything you need to say. You need to tell me what k is. If you just start using k and it wasn’t in the problem, I have no idea what you’re talking about. You need to use units if there are units. You need to name theorems or ideas you’re using. You don’t need to explain everything you know that’s related to the topic.

Show the reader *what your final answer is* and *why it answers the question*. Gymnasts end routines with their arms up and backs arched to show they stuck the landing; musicians and actors end a performance with a bow. You too have to show you’re done!

I’ll end with a two final exams I have given. One of them is a test for people who are working on getting into a master’s program, so don’t worry if you don’t understand the questions — but notice how many questions ask you explain something. The other was a precalculus final from a few years ago. Again… lots of ‘splaining to do.

Keep writing!

]]>In a lot of my college classes I allow* note sheets *on tests, but not for the reasons you think. Just one piece of paper, or one 3×5 notecard — and it’s all a psychological trick.

I allow note sheets — and you should make them — because the simple process of trying to cram all the “useful stuff” onto a little piece of paper makes you think critically about what you’re trying to learn.

“Is this formula important?”

“Can I fit this whole example onto the page? How could I break it down?”

“What’s the pattern for this so I don’t have to draw the whole unit circle?”

**Chunking and thinking about what’s important**

*Chunking *is what researchers call the process of putting together small pieces of information so that you can keep them together. Chunking is part of why it’s easier to remember a sentence like, “The quick brown fox jumped over the lazy dog” in order than, “Sandwich littoral tennis grasping it’s and fang accident to” — same number of words. We put a structure and a meaning on the first sentence, and maybe a picture, and it’s not so hard to remember. You can do that with math learning too: for max-min problems, you have a story of what you should do; for trig problems, you have a picture of what you want to think about. When you’re trying to compress your knowledge onto a piece of paper, you start chunking along the way and you actually learn a lot!

You’re also evaluating the big picture, looking at everything you’ve learned and trying to figure out what is important, what’s at the base of the subject and the test questions, what smaller ideas depend on. This thinking about thinking is also really important for putting it all in context and helping it stick.

**Does a note sheet really help on a test?**

Yes and no.

Yes when it helps you prevent typos in a formula you don’t quite remember. Yes when you wrote down an example you mostly understand (but not quite) and a similar question comes up on the test. Yes when it reminds you to put +C!

No, because you did the learning when you put together the note sheet. I’ve seen people totally forget their note sheet and ace the test — it was doing the chunking and metacognition that was useful. No if you’re using someone else’s sheet, like one printed off the internet! You didn’t do the work to put it together, so you don’t know where to look for the information you want, and you don’t remember how to use it anyway.

**How do I make a good note sheet, then?**

More later!

]]>A bicyclist sets off for school. She takes fifteen minutes to get up to cruising speed, 16 miles per hour. After fifteen minutes of cruising, she remembers she forgot her presentation for class at home: time to turn around! She turns around almost instantaneously and her velocity is suddenly 16 miles per hour in the other direction. She keeps going at 16 miles per hour until she gets home.

(a) Draw a graph of the bicyclist’s velocity on the following axes. Label your units.

(b) Use geometry or Riemann sums to estimate the distance the bicyclist traveled in her first 45 minutes. Show your work clearly.

(This is from a real exam I gave at St. Olaf College about four years ago.)

She had a quick question about something so I got to look at her work — and I was horrified to see that she’d flipped the miles and the minutes upside down and mixed up minutes and hours. Her answer was off by a factor of 1000 or something crazy like that. But the integral and the rest of the process was fine! I felt so bad for her — but it was a test, and I couldn’t say anything.

**Units can save you from silly conversion mistakes. **In any application problem like this, make a habit to write down the units next to every number you’re using. Speed: 16 — is that 16 minutes per hour or miles per hour? WRITE IT DOWN! 16 m/h is 16 miles / 60 minutes. Writing down the units will remind you to convert things: you’ll save yourself from writing 15*16 = 240 so the biker went 240 miles before she even turned around…. ?!

**Units can help you figure out what to do. **You need velocity. You need distance. You have a shaky grasp on the whole calculus thing but you know you need to do an integral and a derivative. Velocity is miles per hour with a sign (unlike speed) and distance is just miles. Miles are…. miles per hour times hours. Hm. In a Riemann sum you’re going to multiply miles per hour by time (hopefully hours!) — that will give you miles. Ok! Check! For part (a) you’ll draw the speed with a sign, and then for part (b) you’ll draw boxes and triangles to find the area between the curve and the horizontal axis.

This could help in the other direction, too. If you have distance f(t), then the derivative of f(t) with respect to t is d f(t) / dt — (instantaneous) change in distance over change in time. That’s miles per hour, so velocity is the derivative of the distance function.

You can even figure out crazy physics formulas just by making sure the units match up. You have some nonsense problem that requires you figure out something in Newtons? Well, Newtons are kilograms times meters per second squared. I bet they gave you a mass (kilograms) and an acceleration (meters per second squared). There’s an 80% chance you’ll get this problem right just by multiplying those numbers, because it’ll make the right units for the answer.

Start this habit right away in precalculus if it’s not too late. It will help so much!

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